[Paper Review] Improved Bounds for Beacon-Based Coverage and Routing in Simple Rectilinear Polygons

Sang Won Bae, Chan-Su Shin+1 more|arXiv (Cornell University)|May 19, 2015

Computational Geometry and Mesh Generation13 references3 citations

TL;DR

This paper establishes tight combinatorial bounds for beacon-based coverage and routing in simple rectilinear polygons. It proves that ⌊n/6⌋ beacons are both necessary and sufficient for full coverage, and ⌊(3n−4)/8⌋−1 beacons are sufficient for routing between any two points, with ⌈n/4⌉−1 beacons sometimes required, significantly improving prior bounds. An optimal O(n)-time algorithm computes the beacon-based kernel.

ABSTRACT

We establish tight bounds for beacon-based coverage problems, and improve the bounds for beacon-based routing problems in simple rectilinear polygons. Specifically, we show that $\lfloor \frac{n}{6} floor$ beacons are always sufficient and sometimes necessary to cover a simple rectilinear polygon $P$ with $n$ vertices. We also prove tight bounds for the case where $P$ is monotone, and we present an optimal linear-time algorithm that computes the beacon-based kernel of $P$. For the routing problem, we show that $\lfloor \frac{3n-4}{8} floor - 1$ beacons are always sufficient, and $\lceil \frac{n}{4} ceil-1$ beacons are sometimes necessary to route between all pairs of points in $P$.

Motivation & Objective

• To close the gap between known lower and upper bounds for beacon-based coverage and routing in simple rectilinear polygons.
• To establish tight, matching bounds for the number of beacons required to cover any simple rectilinear polygon with n vertices.
• To improve the upper and lower bounds for beacon-based routing, reducing the gap from previous results.
• To present an optimal O(n)-time algorithm for computing the beacon-based kernel of a simple rectilinear polygon.
• To address open problems in beacon-based geometry, including the inverse kernel computation.

Proposed method

• Constructs a lower-bound polygonal instance requiring ⌊n/6⌋ beacons for coverage, proving the bound is tight.
• Proposes a recursive partitioning strategy based on reflex edges and cuts to place beacons efficiently in subpolygons.
• Uses geometric properties such as visibility, monotonicity, and convex edge routing to ensure beacon attraction between points.
• Applies a divide-and-conquer approach with beacon placement at strategic vertices (e.g., convex, reflex, cut endpoints) to maintain routing connectivity.
• Employs a recursive beacon placement algorithm that bounds the total number of beacons using the formula ⌊(3r)/4⌋ for subpolygons with r reflex vertices.
• Designs a linear-time algorithm to compute the beacon-based kernel by leveraging monotonicity and visibility structures in rectilinear polygons.

Experimental results

Research questions

• RQ1 What is the tight lower bound on the number of beacons required to cover any simple rectilinear polygon with n vertices?
• RQ2 Can the upper bound for beacon-based coverage be reduced to match the new lower bound?
• RQ3 What is the tightest possible upper bound for beacon-based routing between all pairs of points in a simple rectilinear polygon?
• RQ4 Is there a linear-time algorithm to compute the beacon-based kernel of a simple rectilinear polygon?
• RQ5 Can the inverse kernel (set of points attracted to all points in P) be computed in subquadratic time?

Key findings

• ⌊n/6⌋ beacons are both necessary and sufficient to cover any simple rectilinear polygon with n vertices, establishing a tight bound.
• The paper presents a linear-time algorithm to compute the beacon-based kernel of a simple rectilinear polygon, which is optimal.
• ⌊(3n−4)/8⌋−1 beacons are sufficient for beacon-based routing between all pairs of points in a simple rectilinear polygon.
• ⌈n/4⌉−1 beacons are sometimes necessary for routing, improving the previous lower bound and closing the gap with the new upper bound.
• The routing algorithm ensures that any pair of points can be routed via a sequence of beacons, with connectivity maintained through strategic beacon placement at convex and reflex vertices.
• The results resolve open questions on coverage bounds and improve the state-of-the-art for routing bounds in rectilinear polygons.